Research: Dice Toss Simulation — Hot, Cold & Independence

Generated: 2025-12-15T20:18:04.166Z · Lottery: Dice Toss (1..6)

Pre-registration

Primary hypothesis: Under independence, conditioning on a lagging first block does not make the target appear sooner. (Δ = 0).
Primary metric: Primary outcome = ΔmeanWaitToFirstHit (RMST@200) for lagging − control; one-sided direction = lagging sooner (Δ<0).
Analysis plan: We report analytic 95% CIs for ΔRMST@200 (primary), plus secondary ΔmeanCount@10 and a bounded ΔpHitWithin(X) using a scenario-specific X chosen to avoid ceiling/floor (baseline near ~0.7). We also show a robustness grid over (cutoff, window, alpha).

Design

Mode: target
Rules: main 1 of 6
Seed: 1; Future draws: 200; Grid trials per cell: 2000
Primary horizon (RMST): 200; Secondary meanCount window: 10
Grid: cutoff ∈ {50, 100, 200}, windows ∈ {5, 10, 20, 50, 100}, α ∈ {0.01, 0.05}
UX pHitWithin window: 7
Target mode: random

Primary result (pre-registered)

Mode: target; cutoff=100; alpha=0.05; window=10; RMST horizon=200

Primary outcome: ΔmeanWaitToFirstHit (RMST@200) lagging − control

Definition: draw index of first hit in the future window (1..T), with “no hit” treated as right-censored at T; reported as RMST = E[min(Tfirst, T)].

Estimate: -1.099

95% CI: -2.463 to 0.266

One-sided p (approx): 0.057309

n: lagging=47; control=1908

Note: p-value is one-sided because direction (lagging sooner ⇒ Δ<0) is pre-registered; CIs are two-sided 95%.

Censoring rate: lagging=0.000000; control=0.000000

Secondary outcome: ΔmeanCount in next 10 draws (lagging − control)

Estimate: -0.083

95% CI: -0.411 to 0.246

(Linear expectation metric; not bounded by 0..1.)

Tertiary (bounded / UX metric): ΔpHitWithin(7) lagging − control

Window is chosen to avoid ceiling/floor when possible; still bounded and non-linear.

Estimate: 0.079999

95% CI: -0.034249 to 0.194248

Kaplan–Meier survival curve: probability the target has nothit yet by draw t

Curves should overlap under independence; systematic separation would indicate a real shift in time-to-first-hit.

Interpretation: Lagging numbers do not reach their first hit sooner than comparable non-lagging numbers under independence. Shaded areas represent pointwise 95% confidence intervals.

Limitations

Simulation-based
Assumes independence
Not testing real-world rigging

Data verdict (plain language)

Verdict: No evidence of a ‘due’ effect.

ΔRMST@200 = -1.099 (95% CI -2.463 to 0.266); one-sided p≈0.057309. Estimate is slightly sooner, but not reliably different from 0.
Interpretation: “Lagging sooner” means fewer expected draws until first hit (negative ΔRMST).

Power / sensitivity (approx)

Approximate minimal detectable effect (80% power; normal approximation; two-sided alpha). Approximate also because the cohort is defined by a binomial-tail conditioning event.

Primary metric baseline meanWait (RMST@200): 5.843

n: lagging=47; control=1908

MDE @80% power(|ΔmeanWait|): 1.950

Secondary baseline meanCount@10: 1.700

MDE @80% power(|ΔmeanCount|): 0.469

Interpretation: effects smaller than the MDE are hard to reliably detect with this cohort selection + sample size.

“Approximate” reflects both the normal approximation and the binomial-tail conditioning used to define cohorts.

Tertiary metric baseline: pHitWithin(7) = 0.728512 · MDE @80% power (|ΔpHit|) = 0.183860

Interpretation notes (important)

Independence null: for toy processes (coin/die) the true effect is 0 by construction; any single “significant” result can occur by chance when you scan many cells.
Ceiling/floor risk: when baseline pHitWithin is near 1 (ceiling) or near 0 (floor), ΔpHitWithin is a bounded, non-linear metric and can look more dramatic than it is.
Preferred quantity: the primary endpoint here is mean wait-to-first-hit (RMST@T), which directly targets “does it show up sooner?” and avoids the ceiling pathology of pHitWithin in high-p regimes.
Small cohorts: grid cells with very small cohort sizes (n <20) are shown for completeness but are not interpretable (often producing degenerate CIs like 0 to 0).

What these results do not show

No “compensation” mechanism: a cold streak does not create a forward advantage under independence.
No exploitable strategy: statistically significant cells (especially under ceiling/floor metrics or small n) do not imply predictability or an edge.
No jackpot implication: this does not change the combinatorial odds of a specific full ticket.
No claim about real-world fairness: real lotteries can be tested for bias separately; this report’s main claim is about conditional reasoning under the null.

Robustness grid (lagging vs control)

Each row is one (cutoff, α, window). Look for systematic drift away from 0; do not over-interpret isolated “significant” rows.
Rows with n(lag) or n(ctrl) < 20 are visually muted and are not interpretable.

mode	cutoff	alpha	window	n(lag)	n(ctrl)	Δmean	95% CI (Δmean)	ΔRMST	95% CI (ΔRMST)	ΔpHit	95% CI (ΔpHit)
target	50	0.01	5	2	1996	-0.325	-1.305 to 0.656	-0.111	-3.052 to 2.830	NA	NA
target	50	0.01	10	2	1996	-1.147	-2.128 to -0.165	0.890	-6.951 to 8.732	NA	NA
target	50	0.01	20	2	1996	-0.797	-1.779 to 0.186	0.433	-8.390 to 9.256	NA	NA
target	50	0.01	50	2	1996	-1.211	-1.327 to -1.095	0.250	-8.574 to 9.074	NA	NA
target	50	0.01	100	2	1996	-2.020	-3.012 to -1.027	0.250	-8.574 to 9.074	NA	NA
target	50	0.05	5	43	1930	-0.181	-0.451 to 0.090	0.286	-0.173 to 0.744	-0.157621	-0.307657 to -0.007585
target	50	0.05	10	43	1930	-0.303	-0.664 to 0.057	0.785	-0.231 to 1.801	-0.034173	-0.156944 to 0.088598
target	50	0.05	20	43	1930	-0.279	-0.832 to 0.273	1.018	-0.619 to 2.656	-0.015942	-0.079353 to 0.047470
target	50	0.05	50	43	1930	-0.021	-0.863 to 0.820	1.138	-0.778 to 3.054	0.000000	≈0.000000 (degenerate CI)
target	50	0.05	100	43	1930	-0.469	-1.721 to 0.783	1.138	-0.778 to 3.054	0.000000	≈0.000000 (degenerate CI)
target	100	0.01	5	8	1985	-0.111	-0.602 to 0.380	0.706	-0.014 to 1.427	NA	NA
target	100	0.01	10	8	1985	-0.576	-1.023 to -0.129	0.578	-1.316 to 2.472	NA	NA
target	100	0.01	20	8	1985	0.174	-0.657 to 1.005	-0.079	-2.151 to 1.992	NA	NA
target	100	0.01	50	8	1985	0.957	-1.198 to 3.112	-0.191	-2.264 to 1.882	NA	NA
target	100	0.01	100	8	1985	0.909	-2.568 to 4.386	-0.191	-2.264 to 1.882	NA	NA
target	100	0.05	5	47	1908	0.100	-0.118 to 0.318	-0.451	-0.942 to 0.039	0.131473	0.004910 to 0.258036
target	100	0.05	10	47	1908	-0.083	-0.411 to 0.246	-0.863	-1.784 to 0.058	0.018043	-0.078668 to 0.114753
target	100	0.05	20	47	1908	0.300	-0.154 to 0.754	-0.985	-2.347 to 0.377	0.025681	0.018584 to 0.032779
target	100	0.05	50	47	1908	0.255	-0.495 to 1.006	-1.099	-2.463 to 0.266	0.000000	≈0.000000 (degenerate CI)
target	100	0.05	100	47	1908	-0.680	-1.783 to 0.423	-1.099	-2.463 to 0.266	0.000000	≈0.000000 (degenerate CI)
target	200	0.01	5	4	1986	-0.337	-1.318 to 0.644	0.373	-1.588 to 2.334	NA	NA
target	200	0.01	10	4	1986	0.325	-1.062 to 1.712	1.447	-2.352 to 5.245	NA	NA
target	200	0.01	20	4	1986	0.376	-1.096 to 1.848	0.686	-3.115 to 4.487	NA	NA
target	200	0.01	50	4	1986	1.159	-1.437 to 3.754	0.574	-3.229 to 4.376	NA	NA
target	200	0.01	100	4	1986	1.242	-0.725 to 3.208	0.574	-3.229 to 4.376	NA	NA
target	200	0.05	5	35	1935	0.021	-0.273 to 0.315	0.267	-0.251 to 0.785	-0.037874	-0.203260 to 0.127512
target	200	0.05	10	35	1935	0.185	-0.259 to 0.630	0.383	-0.694 to 1.459	-0.009671	-0.135606 to 0.116263
target	200	0.05	20	35	1935	0.437	-0.157 to 1.031	0.591	-1.108 to 2.290	0.021189	0.014772 to 0.027605
target	200	0.05	50	35	1935	0.648	-0.370 to 1.667	0.476	-1.226 to 2.178	0.000000	≈0.000000 (degenerate CI)
target	200	0.05	100	35	1935	0.753	-0.464 to 1.970	0.476	-1.226 to 2.178	0.000000	≈0.000000 (degenerate CI)

How to Cite This Page

Lucky Picks. “Research: Dice Toss Simulation — Hot, Cold & Independence.”
https://luckypicks.io/research/independence-and-lagging-numbers/dice-toss-simulation/
(Accessed December 2025)

For a non-technical explanation of what these results mean for players, see our Hot & Cold Lottery Numbers guide.